Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central...

Full description

Saved in:
Bibliographic Details
Published in:Journal of algebra 2013-08, Vol.388, p.203-218
Main Authors: Liu, Chia-Hsin, Passman, D.S.
Format: Article
Language:English
Subjects:
3-Group
Central unit
Integral group ring
Multiplicative Jordan decomposition
Wedderburn component
Z-group
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in Z[G] cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of the form Cp⋊C3k, with prime p>7 and with the cyclic 3-group acting like a group of order 3, then Z[G] does not have MJD.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2013.04.015